Subgroups — C₃×#68529 #11776863

Order 1152.

395 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4837	C₂	1	2	yes	maximal
2	C₃×C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃²×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4480	C₂	1	2	yes	maximal, complemented
5	C₃×#4457	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1566	—	3	1	no	maximal
7	#68529	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
9	C₂×C₃×D₃×C₈	C₄	1	1	yes	characteristic
10	C₂×C₃×D₃×C₈	C₄	1	1	yes	characteristic
11	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
12	C₃²×C₄⋊C₈	C₂²	1	2	yes	—
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃×Dic₃⋊C₈	—	2	2	no	—
15	C₃×C₈×Dic₃	C₄	1	1	yes	characteristic
16	C₃×C₈×Dic₃	C₄	1	1	yes	characteristic
17	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×#423	—	3	2	no	—
20	C₂×C₃×C₄×C₈	—	3	1	no	—
21	#4837	C₂×C₃	1	2	yes	—
22	C₄×D₃×C₈	C₂×C₃	1	1	yes	characteristic
23	C₃×#334	—	2	1	no	—
24	C₃×#334	—	3	2	no	—
25	C₃×#334	—	3	1	no	—
26	C₃×#334	D₃	1	1	yes	characteristic, complemented
27	C₃×#334	C₂×C₃	1	1	yes	characteristic, complemented
28	#4480	C₂×C₃	1	2	yes	complemented
29	#4457	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×D₃	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄×D₃	Q₈	1	1	yes	characteristic
33	C₂×C₃²×C₈	—	2	2	no	—
34	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
36	C₃²×C₄²	C₂³	1	1	yes	characteristic
37	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄×Dic₃	D₄	1	1	yes	characteristic
39	C₃×C₄×Dic₃	Q₈	1	1	yes	characteristic
40	C₂×C₃×C₃⋊C₈	—	2	2	no	—
41	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₃×ΓL(1,𝔽₂₅)	—	4	4	no	—
44	C₃×D₃×C₈	—	2	2	no	—
45	C₃×D₃×C₈	—	2	2	no	—
46	#1566	—	3	1	no	Sylow
47	C₂×C₃×M₁₆	—	6	2	no	—
48	C₂²×C₃×C₈	—	3	1	no	—
49	C₂²×C₃×C₈	—	3	1	no	—
50	C₂×C₃×C₄²	—	3	1	no	—
51	C₂×ΓL(1,𝔽₂₅)	—	2	2	no	—
52	C₂×D₃×C₈	C₃×C₄	1	1	yes	characteristic
53	C₂×D₃×C₈	C₃×C₄	1	1	yes	characteristic
54	C₄²×D₃	C₂²×C₃	1	1	yes	characteristic
55	C₃×C₄⋊C₈	—	2	2	no	—
56	C₃×C₄⋊C₈	—	3	2	no	—
57	C₃×C₄⋊C₈	—	6	2	no	—
58	C₃×C₄⋊C₈	C₂×D₃	1	2	yes	—
59	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
60	C₃×C₄×C₈	—	2	1	no	—
61	C₃×C₄×C₈	—	3	1	no	—
62	C₃×C₄×C₈	—	3	1	no	—
63	C₃×C₄×C₈	—	3	1	no	—
64	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
65	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
66	Dic₃⋊C₈	—	2	2	no	—
67	C₈×Dic₃	C₃×C₄	1	1	yes	characteristic
68	C₈×Dic₃	C₃×C₄	1	1	yes	characteristic
69	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
70	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
71	C₂²×C₃×D₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
78	C₃×C₄×D₃	—	2	1	no	—
79	C₃×C₄×D₃	—	2	1	no	—
80	C₃×C₄×D₃	—	2	2	no	—
81	C₃×C₄×D₃	SD₁₆	1	4	yes	—
82	C₃²×C₈	—	2	1	no	—
83	C₃²×C₈	—	4	2	no	—
84	C₃²×C₈	M₁₆	1	2	yes	complemented
85	C₃×C₃⋊C₈	—	2	1	no	—
86	C₃×C₃⋊C₈	—	4	2	no	—
87	C₃×C₃⋊C₈	M₁₆	1	2	yes	complemented
88	#423	—	3	2	no	—
89	C₂×C₄×C₈	—	3	1	no	—
90	#334	—	3	2	no	—
91	#334	—	3	1	no	—
92	#334	C₃×D₃	1	1	yes	characteristic, complemented
93	C₂²×C₃×C₄	—	3	1	no	—
94	C₂²×C₃×C₄	—	3	1	no	—
95	C₂²×C₃×C₄	—	3	1	no	—
96	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂×C₄×D₃	C₃×D₄	1	1	yes	characteristic
98	C₂×C₄×D₃	C₃×Q₈	1	1	yes	characteristic
99	C₃×M₁₆	—	12	4	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	2	1	no	—
103	C₂×C₃×C₈	—	2	1	no	—
104	C₂×C₃×C₈	—	3	1	no	—
105	C₂×C₃×C₈	—	3	1	no	—
106	C₂×C₃×C₈	—	4	2	no	—
107	C₂×C₃×C₈	—	6	2	no	—
108	C₂×C₃×C₈	—	6	2	no	—
109	C₂×C₃×C₈	—	6	2	no	—
110	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
111	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
112	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
114	C₃×C₄²	—	2	1	no	—
115	C₃×C₄²	—	3	1	no	—
116	C₃×C₄²	—	3	1	no	—
117	C₃×C₄²	—	3	1	no	—
118	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
119	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
120	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
121	C₄×Dic₃	C₃×D₄	1	1	yes	characteristic
122	C₄×Dic₃	C₃×Q₈	1	1	yes	characteristic
123	C₂×C₃⋊C₈	—	2	2	no	—
124	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
125	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
126	ΓL(1,𝔽₂₅)	—	4	4	no	—
127	D₃×C₈	—	2	2	no	—
128	D₃×C₈	—	2	2	no	—
129	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₃×D₃	—	2	1	no	—
131	C₂×C₃×D₃	—	2	1	no	—
132	C₂×C₃×D₃	#106	1	2	yes	—
133	C₃²×C₄	—	2	1	no	—
134	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
135	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
136	C₃²×C₄	C₂×SD₁₆	1	2	yes	—
137	C₃×Dic₃	—	2	1	no	—
138	C₃×Dic₃	#106	1	2	yes	—
139	C₃×Dic₃	#108	1	2	yes	—
140	C₂×M₁₆	—	6	2	no	—
141	C₂²×C₈	—	3	1	no	—
142	C₂²×C₈	—	3	1	no	—
143	C₂×C₄²	—	3	1	no	—
144	C₄⋊C₈	—	3	2	no	—
145	C₄⋊C₈	—	6	2	no	—
146	C₄⋊C₈	C₂×C₃×D₃	1	2	yes	—
147	C₄×C₈	—	3	1	no	—
148	C₄×C₈	—	3	1	no	—
149	C₄×C₈	—	3	1	no	—
150	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
151	C₂³×C₃	—	3	1	no	—
152	C₂²×D₃	C₃×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	3	1	no	—
157	C₂×C₃×C₄	—	3	1	no	—
158	C₂×C₃×C₄	—	3	4	no	—
159	C₂×C₃×C₄	—	3	1	no	—
160	C₂×C₃×C₄	—	6	2	no	—
161	C₂×C₃×C₄	—	6	1	no	—
162	C₂×C₃×C₄	—	6	1	no	—
163	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
164	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
165	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
169	C₂×Dic₃	C₃×C₄⋊C₄	1	1	yes	characteristic
170	C₂×Dic₃	C₃×C₄⋊C₄	1	1	yes	characteristic
171	C₂×Dic₃	C₃×C₄⋊C₄	1	1	yes	characteristic
172	C₄×D₃	—	2	1	no	—
173	C₄×D₃	—	2	1	no	—
174	C₄×D₃	—	2	2	no	—
175	C₄×D₃	C₃×SD₁₆	1	4	yes	—
176	C₃×C₈	—	2	1	no	—
177	C₃×C₈	—	2	1	no	—
178	C₃×C₈	—	2	2	no	—
179	C₃×C₈	—	3	2	no	—
180	C₃×C₈	—	4	1	no	—
181	C₃×C₈	—	4	2	no	—
182	C₃×C₈	—	4	2	no	—
183	C₃×C₈	—	6	1	no	—
184	C₃×C₈	—	8	2	no	—
185	C₃×C₈	—	12	2	no	—
186	C₃×C₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
187	C₃×C₈	C₃×M₁₆	1	2	yes	complemented
188	C₃⋊C₈	—	2	1	no	—
189	C₃⋊C₈	—	4	2	no	—
190	C₃⋊C₈	C₃×M₁₆	1	2	yes	complemented
191	C₂×C₃²	#423	1	1	yes	characteristic
192	C₂×C₃²	C₂×#106	1	1	yes	characteristic
193	C₂×C₃²	C₂×#108	1	1	yes	characteristic
194	C₃×D₃	—	2	2	no	—
195	C₂²×C₄	—	3	1	no	—
196	C₂²×C₄	—	3	1	no	—
197	C₂²×C₄	—	3	1	no	—
198	M₁₆	—	12	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	3	1	no	—
201	C₂×C₈	—	3	1	no	—
202	C₂×C₈	—	6	2	no	—
203	C₂×C₈	—	6	2	no	—
204	C₂×C₈	—	6	2	no	—
205	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
206	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
207	C₄²	—	3	1	no	—
208	C₄²	—	3	1	no	—
209	C₄²	—	3	1	no	—
210	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	—	3	2	no	—
213	C₂²×C₃	—	6	1	no	—
214	C₂²×C₃	—	6	1	no	—
215	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
216	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
217	C₂×D₃	—	2	1	no	—
218	C₂×D₃	—	2	1	no	—
219	C₂×D₃	C₃×#106	1	2	yes	—
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	3	2	no	—
226	C₃×C₄	—	3	2	no	—
227	C₃×C₄	—	4	1	no	—
228	C₃×C₄	—	6	1	no	—
229	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
230	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
231	C₃×C₄	D₃×SD₁₆	1	2	yes	—
232	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
233	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
234	C₃×C₄	C₂×C₃×SD₁₆	1	2	yes	—
235	Dic₃	—	2	1	no	—
236	Dic₃	C₃×#106	1	2	yes	—
237	Dic₃	C₃×#108	1	2	yes	—
238	C₃²	#1566	1	1	yes	characteristic, complemented, Sylow
239	C₂³	—	3	1	no	—
240	C₂×C₄	—	3	1	no	—
241	C₂×C₄	—	3	1	no	—
242	C₂×C₄	—	3	4	no	—
243	C₂×C₄	—	3	1	no	—
244	C₂×C₄	—	6	2	no	—
245	C₂×C₄	—	6	1	no	—
246	C₂×C₄	—	6	1	no	—
247	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
248	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
249	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
250	C₈	—	2	1	no	—
251	C₈	—	3	2	no	—
252	C₈	—	4	2	no	—
253	C₈	—	6	1	no	—
254	C₈	—	12	2	no	—
255	C₈	C₃×ΓL(1,𝔽₂₅)	1	2	yes	complemented
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	6	2	no	—
260	C₂×C₃	#4837	1	1	yes	characteristic
261	C₂×C₃	D₃×#106	1	1	yes	characteristic
262	C₂×C₃	D₃×#108	1	1	yes	characteristic
263	C₂×C₃	C₃×#423	1	1	yes	characteristic
264	C₂×C₃	C₂×C₃×#106	1	1	yes	characteristic
265	C₂×C₃	C₂×C₃×#108	1	1	yes	characteristic
266	D₃	—	2	2	no	—
267	C₂²	—	3	2	no	—
268	C₂²	—	6	1	no	—
269	C₂²	—	6	1	no	—
270	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	3	2	no	—
273	C₄	—	3	2	no	—
274	C₄	—	6	1	no	—
275	C₄	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
276	C₄	C₃×D₃×M₁₆	1	1	yes	characteristic
277	C₄	C₃×D₃×SD₁₆	1	2	yes	—
278	C₃	—	2	1	no	—
279	C₃	#68529	1	1	yes	characteristic, direct product factor
280	C₃	C₃×#1566	1	1	yes	characteristic, complemented
281	C₂	—	6	2	no	—
282	C₂	C₃×#4837	1	1	yes	characteristic
283	C₂	C₃×D₃×#106	1	1	yes	characteristic
284	C₂	C₃×D₃×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×#334	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₃	C₄⋊C₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×#334	D₃
hypocenter	C₃	C₃×#1566
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×#334

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₃²×#334	C₂	semidirect
4. C₃×#4480	C₂	semidirect
5. C₃×#4457	C₂	semidirect
7. #68529	C₃	direct (N × H)
26. C₃×#334	D₃	semidirect
27. C₃×#334	C₂×C₃	semidirect
28. #4480	C₂×C₃	semidirect
29. #4457	C₂×C₃	semidirect
84. C₃²×C₈	M₁₆	semidirect
87. C₃×C₃⋊C₈	M₁₆	semidirect
92. #334	C₃×D₃	semidirect
186. C₃×C₈	ΓL(1,𝔽₂₅)	semidirect
187. C₃×C₈	C₃×M₁₆	semidirect
190. C₃⋊C₈	C₃×M₁₆	semidirect
238. C₃²	#1566	semidirect
255. C₈	C₃×ΓL(1,𝔽₂₅)	semidirect
279. C₃	#68529	direct (N × H)
280. C₃	C₃×#1566	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#4837	2
C₃×C₄×D₃×C₈	2
C₃²×#334	2
C₃×#4480	2
C₃×#4457	2
C₃×#1566	3
#68529	3

Subgroups — C3×#68529 #11776863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68529 #11776863